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The relationship between non-abelian matrix Toda lattices and Ising-type models of statistical mechanics
Volume 84A, number 7
PHYSICS LETTERS
17 August 1981
THE RELATIONSHIP BETWEEN NON-ABELIAN MATRIX TODA LATIICES AND ISING-TYPE MODELS OF STATISTICAL MECHANICS D.V. CHUDNOVSKY and G.V. CHUDNOVSKY Department of Mathematics, Columbia University, New York, NY 10027, USA Received 5 June 1981
We show that the Ising model hamiltonian is a special case of a matrix Toda lattice hamiltonian with operator variables from a Clifford algebra of dimension equal to the size of the lattice.
The relationship between higher non-abelian Toda flows and different models of statistical mechanics, ineluding the Ising model is establithed here. We will see how the Ismg model appears from a non-abeian version of the Toda lattice [1] with invariant restrictions added from a Clifford algebra. Let us first of all introduce non-abelian lattice flows. These models were proposed for the first time by Polyakov [1~,to be one-dimensional lattice versions of a nonlinear GL(n*) a-model and they are generalizations of the classical scalar Toda lattice: =
e
un—un_i
Ufl4-lUn
e
n’O,±1,±2
1 0 + “n
where)’,,,
=
—‘
—
,uj/~1
~
~
0
/
‘
‘
fl
~
n,i/)i,/1
and the commutation relations between Un,,,, and are the following [2]: —
[‘~n,ab’~~,cd]
rp L
]
U
—
E~4,,ab’Ur,cd] —0, = 6 (6 U 6 U
—
n~ab’ r,cd
—
nr bc n,ad
)
ad n,bc
The system of quantum hamiltonians can be described as a trace of a product of the local transfer matrices:
N
(1)
A non-abelian Toda lattice is a natural generalization of system (1), where e” is substituted by a nonsingular operator U,, defmed on a Hilbert space H. In ref. [2] we proposed, using the S-matrix approach, the description of higher non-abelian Toda lattice flows, commuting with the flow of the non-abelian Toda lattice. The description of the whole family of commuting hamiltonians in achieved using transfer matrices expressed as 2 X 2 matrices with entries which are operators over H. These transfer matrices are products of local transfer matnces of the form: (0) =
tern of quantum hamiltonians, if U? is represented as m X m matrix ~m p ip ~m
HN(O) = tr
[o_N ,~i£n(O)]
where the trace is the sum of all diagonal elements of the matrix in brackets, and there are 2m of them. As a consequence of the S-matrix approach [2,3], the hamiltonians HN(O) are commuting for different 0. One can write down local hamiltonians, if one considers an expansion of log HN(O) in powers of 0: log HN (0)
=
E H.0/. j=1
The hamiltonians 14 are called higher Toda lattice hamiltonians and the first of them have the form
(L~)~ •L~1.One can in fact describe a sys353
Volume 84A, number 7
PHYSICS LETTERS
N H1
In particular, the invariant restrictions B~ 0 give us a
~
=
tr(F,,),
n=i
non-abelian generalization of a lattice system [5]:
N
~ç1),
2)+ ~N tr( ~
~
H2 = n—i tr(1~, with UN+ 1 U -~
•..
~When the variables —C,,C,~1 C,, in (5) are scalar the system is(5) .
,
called the Kac—Moerbeke lattice. It is known that this
n =1
1. The hamiltonianH2 together with the commutation relations given above, implies the non-abelian Toda lattice dynamic equations [9]: ‘U - U-i’ U U~ ~ ~ ‘2’ —
~ n,x
17 August 1981
n Ix
n+1 n
—
—
n—i
fl
‘~ ‘
These equations make sense both classically and quantum mechanically. In ref. [2] another rule defining higher Toda lattice flows was described. According to this thevariables kth higher matrix Toda n, lattice flow in the Todarule, lattice L~, for integer is equivalent to the kth higher matrix coupled nonlinear Schrödinger equation [4] for every pair of potentials
system can be embedded into a scalar Toda lattice flow (and is, in fact, equivalent to it [6]). We show below that the system (5) in particular contains dynamic equations for the lattice (finite or infinite) two-dimensional Ising model. Before proceeding to this connection, we first present Onsager’s definition of the Ising model hamiltonian [7]. Onsager’s operator variables a~,aare re operators thestandard Fock space 2), that built frominthe HN ~ (C Pauli matrices 1
=
a
(0 \l
1 \~ 0/’
a2
=
(0 \_i
a3
i~
oJ
‘
=
(1
0
\o
—l
In particular, the second higher Toda flow is equi-
valent to a coupled nonlinear Schrodinger equation which has the form
The operator variables ah have the following form:
i1,2,3,
—(U~ ) =(u~ n—i
t
fl~i
)
1U,,L~1
(3)
.
XX
The non-abelian Toda lattice equation itself determines all elements U,, of the Toda chain starting with the first two of them: U1 and (J0. The formula here is the following: =
~
—
~
+
1
—
,r
,~—
Bn
=
i
—
Un,x - U~’ n ‘
cn
=
un u—1 n—i
.
Then the non-abelian Toda lattice equations have the form: ~
=
—
C,,,
C,,~= B,,C,,
—
.
In terms of B,, and C,,, we can express the second Toda lattice flow in the form 2C,, C~B~ C~C,,_ C,~ C,,+iC~+B,, 1 354
pends on two coupling constants E1 and E2 and can be written as
N
=
—
~(E~a,,~a~1+E2a,~).
(6)
The evolution of this hamiltonian in time can be described using the Uouville theorem aah/at = [a,’,,~C1] together with the commutation relations satisfied by ~ We show ref F8that the time evolution of (6) is a particular case of eq. (5), when one imposes on the van0flk
•L
ables C,, the condition that thealgebra. C,, obey the relations for generators of the Clifford These constraints on (5) can be written in the following form:
c,~=i, ~
(7)
C~B,,_ 1
—
de-
n=1
)x~’n+i~z n”n—1~ In a more traditional form this equation can be written in terms ofB’s and C’s. Let us put ~fl,Xfl
In these variables the Ising model hamiltonian ~
£ç1U,11.’1U,,,
which is merely a consequence of a non-abelian Toda flow: —1~ —
n1,...,N.
+2&~
(4)
It is easy to check that these constraints, as well as the following additional constraints
C,,2C,,~ if In1 —n21*l are invariant for eq. (5). C,,1C,,2
,
(8)
Volume 84A, number 7
PHYSICS LETTERS
The Ising model corresponds to the realization of these Clifford algebra relations (7), (8), in terms of the 1 and a3: C Pauli matrices a 1 ®a1® ® 2,, = ea ...
...
“
C2n+l
=
...
®~®
~
n+i •..®~®
17 August 1981
tonian (6) (linear in C,,) leads to complete diagonalization of ~C1and representation ofthe Onsager equations as a system of linear equations. We present the corresponding version ofthe Kramers—Wannier transformations,as used by Onsager. In this transformation we substitute the commutation relations (7) and (8) for C~by true Clifford algebra anticommutation relations
....
The non-abelian lattice system (5) together with restrictions (7) and (8) will be called generalized Onsager equations. Though these equations describe a
on fermionic fields: ‘I’,,, x,,. The transformation is as follows: C
2,, = i’4’,,x,~_1 C~,,+1= i X,,’4’,, The commutation relations on ‘I’,,, x,, are the following ones: 26ni ,n2’ {x~i~ Xn2} = 26,, i,,,2’ {‘4’~ 1’ X ‘4’n2} {‘~‘ni’ ,,2} = 0. ,
generalization of the two-dimensional Ising model, we call them Onsager equations because they came not directly from the Ising model hamiltonian, but from aspin hamiltonian of theand Ising model in terms of the operators form proposed investigated by Onsager [7]. In the most general case this hamiltonian form can be expressed in terms of an infinite Clifford alge-
bra. We consider the system of operators C,, which obey the rules (7) and (8) above. Like in the ordinary Ising model, we have two coupling constants. These coupling constants are attached to odd and even integer indices n. If these coupling constants are denoted by Ei and E2 then we define E~to be E1 or E2 depending on whether n is odd or even, respectively. The hamiltonian describing the Onsager equations is the followingone [8]: H1
=
~ E,,
Here ~a, b} = ab + ba is an anticommutator. In these new variables the hamiltonian H 1 becomes quadratic and the corresponding generalized Onsager equations become linear. Indeed, we have
H2
=
-~
~ E1’ x,,’4’,, + ~ E~‘4.’,, x,,
—
The evolution equations are the following: ‘
a’I’,,/at=E1x,, —E2x,,1 aX,,/at=E2’4’,, —E’I’,,~1
These classical Kramers—Wigner transformations had
C,,
This hamiltonian is identical to the Ising model hamiltonian (6) with the identification C2,, = u~a~+ ~ 3 C2,,~1= a,,. Though this hamiltonian is linear in C,,, the evolution equations are nonlinear, since the C~obey nontrivial commutation relations. The evolution equations for the operator variable C,, can be written down using the Liouville theorem: ‘
been applied by Onsager in order to linearize the twodimensional Ising model. However, from the point of view of the Toda lattice we consider, there is no need to introduce separate fermionic variables. The initial variables L~,can be considered as anticommuting operators from a Clifford algebra. It is easy to check that the commutation relations on C,, (7) and (8) are equivalent to the statement that the operator variables U~, are generators of a Clifford algebra:
aC,,Iat=[C,,,Hi] This gives us, taking into account commutation relations (7) and (8) on C,,, the equations C~,~=E,,÷iC,,+iC,, —E,,1C,,C,,.1 These equations coincide with the generalized Onsager equations (5) [with constraints (7) and (8)] if one sub-
stitutes C,, by C,,E,,. The surprisingly simple form of the Onsager hamil-
(9) Of course, in the case of a periodic lattice the indices are understood mod N. It is easy to verify that these constraints on U,, are invariant with respect to all Toda flows.
With the Clifford relations (9) on U,,, the generalized Onsager equations on C,, = U,~ are linearized in terms of the variables L.~,only. The corre355
Volume 84A, number 7
PHYSICS LETTERS
References
sponding linear equations are =
~
—
.
(10)
These equations should be considered simultaneously with anticommutation relations (9) on l~,.In this case it implies the second higher Toda latticeflow (3). Moreover, with the Clifford relations (9) on (4, all
higher Toda lattice flows can be linearized. The linearization of Toda lattice flows using algebraic transformations in the periodic case is impossible (unless special commutation relations are added), because reduction to action—angle variables rests on Abel’s map and taking quadratures of abelian integrals. However, in the case of an “open” Toda lattice, when interaction between UN and U1 is absent, the linearization is possible without any additional restrictions on
In this case the flow is always reduced to a linear one and solutions are always given in terms of elementary functions, (4,.
This research was supported in part by the Office of Naval Research under contracts N0001 4-78-C-01 38 and NRO4I-529.
356
17 August 1981
[1] A.M. Polyakov, Phys. Lett. 82B (1979) 247. [2] D.V. Chudnovsky and G.V. Chudnovsky, Phys. Lett. 82A (1981) 271. [3] R. Baxter, Models of statistical mechanics (Academic Press, 1981), to be published; A.B. Zamolodchikov, Commun. Math. Phys. 69 (1979) 165.
[4] D.V. Chudnovsky, Lecture notes in physics, Vol. 120 (Springer, Berlin, 1980) p. 103; Lecture notes in physics, Vol. 126 (Springer, Berlin, 1980) pp. 352—417. [5] M. Kac and P. van Moerbeke, Adv. Math. 16(1975)160. [6] H.P. McKean, Lecture notes in mathematics, Vol. 675 (Springer, Berlin, 1980). [7] L. Onsager, Phys. Rev. 65 (1944) 117;
C.J. Thompson, Mathematical statistical mechanics (Princeton U.P., 1972). [8] Y. Bashiov and S. Pokrovsky, Commun. Math. Phys. 76 (1980) 129. [9] M. Bruschi, S. Manakov, 0. Ragnisco and D. Levi, to be published.
PHYSICS LETTERS
17 August 1981
THE RELATIONSHIP BETWEEN NON-ABELIAN MATRIX TODA LATIICES AND ISING-TYPE MODELS OF STATISTICAL MECHANICS D.V. CHUDNOVSKY and G.V. CHUDNOVSKY Department of Mathematics, Columbia University, New York, NY 10027, USA Received 5 June 1981
We show that the Ising model hamiltonian is a special case of a matrix Toda lattice hamiltonian with operator variables from a Clifford algebra of dimension equal to the size of the lattice.
The relationship between higher non-abelian Toda flows and different models of statistical mechanics, ineluding the Ising model is establithed here. We will see how the Ismg model appears from a non-abeian version of the Toda lattice [1] with invariant restrictions added from a Clifford algebra. Let us first of all introduce non-abelian lattice flows. These models were proposed for the first time by Polyakov [1~,to be one-dimensional lattice versions of a nonlinear GL(n*) a-model and they are generalizations of the classical scalar Toda lattice: =
e
un—un_i
Ufl4-lUn
e
n’O,±1,±2
1 0 + “n
where)’,,,
=
—‘
—
,uj/~1
~
~
0
/
‘
‘
fl
~
n,i/)i,/1
and the commutation relations between Un,,,, and are the following [2]: —
[‘~n,ab’~~,cd]
rp L
]
U
—
E~4,,ab’Ur,cd] —0, = 6 (6 U 6 U
—
n~ab’ r,cd
—
nr bc n,ad
)
ad n,bc
The system of quantum hamiltonians can be described as a trace of a product of the local transfer matrices:
N
(1)
A non-abelian Toda lattice is a natural generalization of system (1), where e” is substituted by a nonsingular operator U,, defmed on a Hilbert space H. In ref. [2] we proposed, using the S-matrix approach, the description of higher non-abelian Toda lattice flows, commuting with the flow of the non-abelian Toda lattice. The description of the whole family of commuting hamiltonians in achieved using transfer matrices expressed as 2 X 2 matrices with entries which are operators over H. These transfer matrices are products of local transfer matnces of the form: (0) =
tern of quantum hamiltonians, if U? is represented as m X m matrix ~m p ip ~m
HN(O) = tr
[o_N ,~i£n(O)]
where the trace is the sum of all diagonal elements of the matrix in brackets, and there are 2m of them. As a consequence of the S-matrix approach [2,3], the hamiltonians HN(O) are commuting for different 0. One can write down local hamiltonians, if one considers an expansion of log HN(O) in powers of 0: log HN (0)
=
E H.0/. j=1
The hamiltonians 14 are called higher Toda lattice hamiltonians and the first of them have the form
(L~)~ •L~1.One can in fact describe a sys353
Volume 84A, number 7
PHYSICS LETTERS
N H1
In particular, the invariant restrictions B~ 0 give us a
~
=
tr(F,,),
n=i
non-abelian generalization of a lattice system [5]:
N
~ç1),
2)+ ~N tr( ~
~
H2 = n—i tr(1~, with UN+ 1 U -~
•..
~When the variables —C,,C,~1 C,, in (5) are scalar the system is(5) .
,
called the Kac—Moerbeke lattice. It is known that this
n =1
1. The hamiltonianH2 together with the commutation relations given above, implies the non-abelian Toda lattice dynamic equations [9]: ‘U - U-i’ U U~ ~ ~ ‘2’ —
~ n,x
17 August 1981
n Ix
n+1 n
—
—
n—i
fl
‘~ ‘
These equations make sense both classically and quantum mechanically. In ref. [2] another rule defining higher Toda lattice flows was described. According to this thevariables kth higher matrix Toda n, lattice flow in the Todarule, lattice L~, for integer is equivalent to the kth higher matrix coupled nonlinear Schrödinger equation [4] for every pair of potentials
system can be embedded into a scalar Toda lattice flow (and is, in fact, equivalent to it [6]). We show below that the system (5) in particular contains dynamic equations for the lattice (finite or infinite) two-dimensional Ising model. Before proceeding to this connection, we first present Onsager’s definition of the Ising model hamiltonian [7]. Onsager’s operator variables a~,aare re operators thestandard Fock space 2), that built frominthe HN ~ (C Pauli matrices 1
=
a
(0 \l
1 \~ 0/’
a2
=
(0 \_i
a3
i~
oJ
‘
=
(1
0
\o
—l
In particular, the second higher Toda flow is equi-
valent to a coupled nonlinear Schrodinger equation which has the form
The operator variables ah have the following form:
i1,2,3,
—(U~ ) =(u~ n—i
t
fl~i
)
1U,,L~1
(3)
.
XX
The non-abelian Toda lattice equation itself determines all elements U,, of the Toda chain starting with the first two of them: U1 and (J0. The formula here is the following: =
~
—
~
+
1
—
,r
,~—
Bn
=
i
—
Un,x - U~’ n ‘
cn
=
un u—1 n—i
.
Then the non-abelian Toda lattice equations have the form: ~
=
—
C,,,
C,,~= B,,C,,
—
.
In terms of B,, and C,,, we can express the second Toda lattice flow in the form 2C,, C~B~ C~C,,_ C,~ C,,+iC~+B,, 1 354
pends on two coupling constants E1 and E2 and can be written as
N
=
—
~(E~a,,~a~1+E2a,~).
(6)
The evolution of this hamiltonian in time can be described using the Uouville theorem aah/at = [a,’,,~C1] together with the commutation relations satisfied by ~ We show ref F8that the time evolution of (6) is a particular case of eq. (5), when one imposes on the van0flk
•L
ables C,, the condition that thealgebra. C,, obey the relations for generators of the Clifford These constraints on (5) can be written in the following form:
c,~=i, ~
(7)
C~B,,_ 1
—
de-
n=1
)x~’n+i~z n”n—1~ In a more traditional form this equation can be written in terms ofB’s and C’s. Let us put ~fl,Xfl
In these variables the Ising model hamiltonian ~
£ç1U,11.’1U,,,
which is merely a consequence of a non-abelian Toda flow: —1~ —
n1,...,N.
+2&~
(4)
It is easy to check that these constraints, as well as the following additional constraints
C,,2C,,~ if In1 —n21*l are invariant for eq. (5). C,,1C,,2
,
(8)
Volume 84A, number 7
PHYSICS LETTERS
The Ising model corresponds to the realization of these Clifford algebra relations (7), (8), in terms of the 1 and a3: C Pauli matrices a 1 ®a1® ® 2,, = ea ...
...
“
C2n+l
=
...
®~®
~
n+i •..®~®
17 August 1981
tonian (6) (linear in C,,) leads to complete diagonalization of ~C1and representation ofthe Onsager equations as a system of linear equations. We present the corresponding version ofthe Kramers—Wannier transformations,as used by Onsager. In this transformation we substitute the commutation relations (7) and (8) for C~by true Clifford algebra anticommutation relations
....
The non-abelian lattice system (5) together with restrictions (7) and (8) will be called generalized Onsager equations. Though these equations describe a
on fermionic fields: ‘I’,,, x,,. The transformation is as follows: C
2,, = i’4’,,x,~_1 C~,,+1= i X,,’4’,, The commutation relations on ‘I’,,, x,, are the following ones: 26ni ,n2’ {x~i~ Xn2} = 26,, i,,,2’ {‘4’~ 1’ X ‘4’n2} {‘~‘ni’ ,,2} = 0. ,
generalization of the two-dimensional Ising model, we call them Onsager equations because they came not directly from the Ising model hamiltonian, but from aspin hamiltonian of theand Ising model in terms of the operators form proposed investigated by Onsager [7]. In the most general case this hamiltonian form can be expressed in terms of an infinite Clifford alge-
bra. We consider the system of operators C,, which obey the rules (7) and (8) above. Like in the ordinary Ising model, we have two coupling constants. These coupling constants are attached to odd and even integer indices n. If these coupling constants are denoted by Ei and E2 then we define E~to be E1 or E2 depending on whether n is odd or even, respectively. The hamiltonian describing the Onsager equations is the followingone [8]: H1
=
~ E,,
Here ~a, b} = ab + ba is an anticommutator. In these new variables the hamiltonian H 1 becomes quadratic and the corresponding generalized Onsager equations become linear. Indeed, we have
H2
=
-~
~ E1’ x,,’4’,, + ~ E~‘4.’,, x,,
—
The evolution equations are the following: ‘
a’I’,,/at=E1x,, —E2x,,1 aX,,/at=E2’4’,, —E’I’,,~1
These classical Kramers—Wigner transformations had
C,,
This hamiltonian is identical to the Ising model hamiltonian (6) with the identification C2,, = u~a~+ ~ 3 C2,,~1= a,,. Though this hamiltonian is linear in C,,, the evolution equations are nonlinear, since the C~obey nontrivial commutation relations. The evolution equations for the operator variable C,, can be written down using the Liouville theorem: ‘
been applied by Onsager in order to linearize the twodimensional Ising model. However, from the point of view of the Toda lattice we consider, there is no need to introduce separate fermionic variables. The initial variables L~,can be considered as anticommuting operators from a Clifford algebra. It is easy to check that the commutation relations on C,, (7) and (8) are equivalent to the statement that the operator variables U~, are generators of a Clifford algebra:
aC,,Iat=[C,,,Hi] This gives us, taking into account commutation relations (7) and (8) on C,,, the equations C~,~=E,,÷iC,,+iC,, —E,,1C,,C,,.1 These equations coincide with the generalized Onsager equations (5) [with constraints (7) and (8)] if one sub-
stitutes C,, by C,,E,,. The surprisingly simple form of the Onsager hamil-
(9) Of course, in the case of a periodic lattice the indices are understood mod N. It is easy to verify that these constraints on U,, are invariant with respect to all Toda flows.
With the Clifford relations (9) on U,,, the generalized Onsager equations on C,, = U,~ are linearized in terms of the variables L.~,only. The corre355
Volume 84A, number 7
PHYSICS LETTERS
References
sponding linear equations are =
~
—
.
(10)
These equations should be considered simultaneously with anticommutation relations (9) on l~,.In this case it implies the second higher Toda latticeflow (3). Moreover, with the Clifford relations (9) on (4, all
higher Toda lattice flows can be linearized. The linearization of Toda lattice flows using algebraic transformations in the periodic case is impossible (unless special commutation relations are added), because reduction to action—angle variables rests on Abel’s map and taking quadratures of abelian integrals. However, in the case of an “open” Toda lattice, when interaction between UN and U1 is absent, the linearization is possible without any additional restrictions on
In this case the flow is always reduced to a linear one and solutions are always given in terms of elementary functions, (4,.
This research was supported in part by the Office of Naval Research under contracts N0001 4-78-C-01 38 and NRO4I-529.
356
17 August 1981
[1] A.M. Polyakov, Phys. Lett. 82B (1979) 247. [2] D.V. Chudnovsky and G.V. Chudnovsky, Phys. Lett. 82A (1981) 271. [3] R. Baxter, Models of statistical mechanics (Academic Press, 1981), to be published; A.B. Zamolodchikov, Commun. Math. Phys. 69 (1979) 165.
[4] D.V. Chudnovsky, Lecture notes in physics, Vol. 120 (Springer, Berlin, 1980) p. 103; Lecture notes in physics, Vol. 126 (Springer, Berlin, 1980) pp. 352—417. [5] M. Kac and P. van Moerbeke, Adv. Math. 16(1975)160. [6] H.P. McKean, Lecture notes in mathematics, Vol. 675 (Springer, Berlin, 1980). [7] L. Onsager, Phys. Rev. 65 (1944) 117;
C.J. Thompson, Mathematical statistical mechanics (Princeton U.P., 1972). [8] Y. Bashiov and S. Pokrovsky, Commun. Math. Phys. 76 (1980) 129. [9] M. Bruschi, S. Manakov, 0. Ragnisco and D. Levi, to be published.